Questions on (ordinary) differential equations. For questions specifically concerning partial differential equations, use the (pde) tag.

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0
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1answer
44 views

Derivative of a differential equation help??

Please can someone explain this to me in detail: if $y''+4y'+3y=14\cos(2x)$ and $z'''+4z''+3z'=-28\sin(2x)$ show that the $z=y+c$ where $c$ is a constant I know the second is the integral of the first ...
0
votes
1answer
494 views

Finding the interval where a solution is certain to exist for the equation $y' + (\tan t)y = \sin t$

Given the following problem: Determine (without solving the problem) an interval in which the solution is certain to exist for the initial value problem $y' + (\tan t)y = \sin t, \space y(2\pi) = ...
0
votes
1answer
80 views

Find the solution for a boundary value problem

Please, how can we find the solution of this second order boundary value problem $$-(e^{-2x}u')'-\ln(x^2+2)u= 2 e^ {-2x} - x \ln(x^2+2),\,\, x\in ]0,1[, u(0)=0,u(1)=1?$$ Or more generally, What's the ...
0
votes
1answer
38 views

Solve a differential equation and evaluate the solution at a particular value of independent variable

If $\frac{dy(x)}{dx}=(2-3i)y(x)$ where $i=\sqrt{-1}$, what is the value of $y(\pi)$?
0
votes
2answers
77 views

Differential Equations/IVP: $\frac{dy}{dt} = 4 - y^3$ and $y(-1)=2$

Transform the given initial value problem into an equivalent problem with the initial point at the origin. $$\cfrac {dy}{dt} = 4 - y^3 \\ y(-1)=2$$ I have no idea about how to solve it. Could you ...
0
votes
3answers
577 views

Integrating absolute value function

I'm working on a problem drawing phase plane diagrams in my applied mathematics course. I'm supposed to draw the phase line diagram of $x''+\vert x\vert=0.$ In the process, I get to the differential ...
0
votes
1answer
321 views

Wronskian-Differential Equations

The equations below are matrices: Consider the vectors $y^{(1)} (t)$=$\begin{pmatrix}t \\1 \end{pmatrix}$ and $y^{(2)}$ (t)=$\begin{pmatrix}t^2 \\2t \end{pmatrix}$ (a) Compute the Wronskian of ...
0
votes
1answer
82 views

Let $f$ be a field with only one singularity in the origin. Show that the phase diagram of the field $f$ has exactly three distinct orbits

Let $f:\mathbb R\to \mathbb R $ be a field with only one singularity in the origin. Show that the phase diagram of the field $f$ has exactly three distinct orbits which are the following: I need ...
0
votes
2answers
322 views

Solve separable DE with integrating factor and homogeneous substitution

I just came out of test which asked to solve $$\frac{dy}{dx}=\frac{y}{x}$$ with $x,y>0$ in three ways: by separating the variables, using the substitution $y=vx$ and using an integrating factor. ...
0
votes
2answers
460 views

Find a particular solution of the differential equation $-3y''-2y'+y=3xe^x$

Using the method of undetermined coefficients. Guess $(Ax+B)e^x$ Plug into diff eq: $-3[(Ax+B)e^x]'' - 2[(Ax+B)e^x]' + (Ax+B)e^x = 3xe^x$ Wolfram alpha simplifies this to: $A(x-2)=e^x(4B+3x)$. ...
-1
votes
1answer
68 views

Initial values are lost (diff eq to Transfer function)?

I read eternal Julius O. Smith III and he says that $$x_{n-m} = z^{-m}X(z)$$ Particularly, difference relation $$y_{n} = y_{n-1} + x_{n}$$ is solved by by $$Y = z^{-1}Y + X = {X \over ...
-2
votes
1answer
41 views

Differential Equations: solve the system

Solve the following system: $$dx/dt=-.2(y-2)$$ $$dy/dt=.8(x-2)$$ This is what I have so far, but I got stuck.. $$\begin{eqnarray} dx/dt&=&-.2y-.4\\ x'&=&-.2y-.4\\ ...
-6
votes
1answer
280 views

$\frac{dy}{dx} = 3y^{2/3}$ general solution?

What's the general solution of $\frac{dy}{dx} = 3y^{2/3}$ ? Im pretty sure this is a separable equation, but I'm not sure how to go forward? Just multiply by $dx$ and $\frac{1}{3y^{2/3}}$ well then I ...
32
votes
2answers
2k views

Why are mathematician so interested to find theory for solving partial differential equations but not for integral equation?

Why are mathematician so interested to find theory for solving partial differential equations (for example Navier-Stokes equation) but not for integral equations?
25
votes
0answers
217 views

Solving Special Function Equations Using Lie Symmetries

The lie group + representation theory approach to special functions & how they solve the ode's arising in physics is absolutely amazing. I've given an example of it's power below on Bessel's ...
21
votes
8answers
2k views

What's so special about sine? (Concerning $y'' = -y$)

In an attempt to actually grok sine, I came across the $y''= -y$ definition. This is incredibly cool, but it leads me to a whole new series of questions. Sine seems pretty prevalent ...
11
votes
1answer
495 views

Osgood condition

Let $h$ and $g$ be continuous, non-decreasing and concave functions in the interval $[0,\infty)$ with $h(0)=g(0)=0$ and $h(x)>0$ and $g(x)>0$ for $x>0$ such that both satisfy the Osgood ...
14
votes
3answers
1k views

Solving Differential Functional Equation $f(2x)=2f(x)f'(x)$

Find all functions satisfying $f(2x)=2f'(x)f(x)$ Under given condition, can't we find explicit solutions?
12
votes
4answers
429 views

Why solutions of $y''+(w^2+b(t))y=0$ behave like solutions of $y''+w^2y=0$ at infinity

Assume $w>0$ and $b(t)$ be continuous on $[0,+\infty)$ and $\int_0^\infty |b(t)| dt <\infty$ show that $y''+(w^2+b(t))y=0$ has solution $\phi(t)$ such that $$\lim_{t\to\infty} ...
4
votes
1answer
2k views

Locally or Globally Lipschitz-functions

Determine if the following function satisfies a local or a a uniform Lipschitz condition. The definition of locally Lipschitz and globally lipschitz are as follows: (i) We say that f is (uniformly) ...
13
votes
3answers
659 views

When do the Freshman's dream product and quotient rules for differentiation hold?

This is motivated by looking at the calculus exams of some of my undergraduate students. A recurring mistake is assuming that the derivative of the product of functions is a product of derivatives and ...
10
votes
1answer
640 views

What is, how do you use, and why do you use differentials? What are their practical uses?

What is a differential? And how is it useful? What is its practical use? For example, in Electromagnetic Wave Theory as it pertains to diffraction gratings, we have an equation like this one: ...
3
votes
3answers
474 views

Numerical Analysis References

Could anyone suggest any good (perhaps online ref papers) reference material on numerical analysis focusing on determining accuracy/estimated errors, rates/orders of convergence especially when ...
13
votes
1answer
512 views

Tough Inverse Fourier Transform

In reference to this answer I gave the other day, I came across a very interesting function whose IFT would be nice to evaluate as part of completing the solution to the problem I answered. The ...
9
votes
1answer
562 views

Complicated exercise on ODE

I have this exercise extracted from a examination of qualitative theory of ODE (in which we study the existence and uniqueness of solutions, and stability using the function of Lyapunov) I don't know ...
7
votes
1answer
189 views

Properties of the solutions to $x'=t-x^2$

Let $f_c$ be the solution to $$ \left\{ \begin{array}{c} x'=t-x^2 \\ x(0) =c \end{array} \right. $$ I'm trying to prove: If $c \geq 0$ then $f_c(t)$ is defined for all $t>0$ There is a ...
7
votes
1answer
110 views

dropping a particle into a vector field, part 2

Okay, so earlier I posted this question "dropping a particle into a vector field " as sort of a feeler question as i study line integrals in order to go into surface integrals and eventually ...
7
votes
4answers
1k views

Methods to solve differential equations

We are given the equation $$\frac{1}{f(x)} \cdot \frac{d\left(f(x)\right)}{dx} = x^3.$$ To solve it, "multiply by $dx$" and integrate: $\frac{x^4}{4} + C = \ln \left( f(x) \right)$ But $dx$ is not a ...
4
votes
3answers
201 views

How $\frac{dx}{dy}=f(x)g(y) \Leftrightarrow \int \frac{dx}{f(x)} = \int g(y)dy$?

In my intro differential equations class we have often used the "equivalence" stated in title. It seems to me that somehow, the intermediate step $$ \frac{dx}{f(x)} = g(y)dy$$ is being used, in which ...
3
votes
2answers
117 views

Using the Jordan form Complex

Let $C$ be a complex $n \times n$ matrix with $\det C \neq 0$. Prove that there is a complex matrix $B$ such that $C = e^B$ Hint: use the Jordan form matrices for comlexas
2
votes
1answer
425 views

Cancelling differentials

I'll start with an example. In physics, $x(t)$ represents the $x$-position of a particle, and $v(t)$ its ($x$-)velocity. To determine the total displacement of a particle on the interval $[a, b]$, we ...
14
votes
1answer
533 views

How to make a smart guess for this ODE

I am dealing with a strange problem currently, we have a differential equation $$y(x)^2 = \pm \sqrt{-A \cos(x) - B \cos^2(x)+y'(x)-C},$$ where $C, A$ and $B $ are parameters. (The case that either ...
12
votes
4answers
1k views

Examples of nonlinear ordinary differential equations with elementary solutions.

I am looking for nice examples of nonlinear ordinary differential equations that have simple solutions in terms of elementary functions. (But are not trivial to find, like, for example, with ...
11
votes
4answers
356 views

How to prove that $\frac{d^n}{dx^n}(x^2-1)^n=0$ has $n$ real roots?

How do I prove that $$\frac{d^n}{dx^n}(x^2-1)^n=0$$ has $n$ real roots?
8
votes
1answer
701 views

Eigenfunctions of the Laplacian

I am willing to offer a bounty for this one, so I will give you an exact idea of what I need: I am looking for solutions of $$\Delta \Psi(r,\theta)=k^2\Psi(r,\theta)$$ where $k\in \mathbb{R}$. Such ...
7
votes
2answers
366 views

A proof of a theorem of Liouville

I need some reference for the proof of the following theorem attributed to Liouville: Theorem. Let $f(x):\Omega\longrightarrow \mathbb R^n$ be a $C^2$ function where $\Omega$ is an open subset of ...
6
votes
2answers
289 views

Proving Nonhomogeneous ODE is Bounded

I am trying to prove the following: Let $x(t)$ be a solution of the IVP $$ \dot x=A(t)x+h(t), $$ where $A(t), h(t)$ continuous on $1\le t<\infty$. Further assume that $$ \int_1^\infty \| ...
6
votes
1answer
208 views

What is the physical meaning of fractional calculus?

What is the physical meaning of the fractional integral and fractional derivative? And many researchers deal with the fractional boundary value problems, and what is the physical background? What ...
6
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2answers
333 views

Determinant called Grammian

Famously, if functions $f_1,f_2,…,f_n$, each of which possesses a derivative of order $n-1$, are linearly independent on the interval $I$, if $$ \det\left( \begin{array}{ccccc} f_1 & f_2 & ...
6
votes
3answers
228 views

Solutions of homogeneous linear differential equations are a special case of structure theorem for f.g. modules over a PID

In this other post, Qiaochu Yuan comments that the solutions for the homegeneous linear differential equation with constant coefficients are a special case of the structure theorem for finitely ...
5
votes
1answer
86 views

Decomposite a vector field into two parts

Let A be a region in $\mathbb R^3$, and suppose $ \vec {\mathbf F}$ is a smooth vector field on A. I was asked to show that I can write $\vec {\mathbf F}=\vec {\mathbf F_1}+\vec {\mathbf F_2}$, s.t. ...
4
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6answers
1k views

ODE introduction textbook

Unfortunately I have reached the maximum number of math classes I can take for my undergraduate degree. I still wish to study basic ODEs and basic number theory. What is a good textbook with an ...
3
votes
1answer
39 views

Uniqueness of solutions to linear recurrence relations

I understand that if I have a linear homogeneous recurrence relation of the form $q_n = c_1 q_{n-1} + c_2 q_{n-2} + \cdots + c_d q_{n-d}$, I can construct the characteristic polynomial $p(t) = t^d - ...
3
votes
2answers
389 views

Advection Diffusion Equation on Semi-Infinite Domain

Regarding the BVP $$u_t(x,t) - v u_x(x,t)=k u_{xx}(x,t),\qquad x\geq0$$ with BC $u_x(0, t)=0$ for $t\geq 0$, and parameters $v,k>0$, I have some questions. Does an expression for the Green's ...
2
votes
2answers
441 views

How to solve this ODE?

For a certain problem, I am trying to solve the ODE $$\ddot{z}(t) - \omega^2 z(t) = f_0 \Big(e^{-i(\omega+\delta)t}+e^{-i(\omega-\delta)t}\Big)$$ where $\omega$ is a real and $\delta$ is close to ...
1
vote
2answers
89 views

Suggestion for a Lyapunov function

Consider the differential system $$ x'=x+y $$ $$y'=x-y+xy$$ What would be a Lyapunov function for this system at $(0,0)$? I have considered functions $V(x,y)=ax^{2n}+by^{2m}$ but none of ...
1
vote
1answer
2k views

Pursuit Curve. Dog Chases Rabbit. Calculus 4.

(a) In Example 1.21, assume that $a$ is less than $b$ (so that $k$ is less than $1$) and find $y$ as a function of $x$. How far does the rabbit run before the dog catches him? (b) Assume now that ...
0
votes
1answer
113 views

Questions about the hyperbolic system of equations

$$u_t+A(x,t,u)u_x=b(x,t,u) \tag 1$$ $$u=(u_1, \dots, u_n), b=(b_1, \dots, b_n)$$ $$A=[a_{ij}], i,j = 1, \dots, n$$ $$$$ We set the question if there are characteristic directions at the path of which ...
8
votes
2answers
208 views

A Differential operator.

What are the fundamental solutions for the operator $$\mathcal D=a{\partial^2\over\partial x_1^2}+b{\partial^2\over\partial x_2^2}$$ on $\Bbb R^2 $ with standard cordinates $(x_1,x_2)$. Here ...
8
votes
1answer
410 views

To get addition formula of $\tan (x)$ via analytic methods

Assume that we only know $\tan (0)=0$ and also given the relation $\tan'(x)=1+\tan^2(x)$ about $\tan (x)$ and we do not know other $\tan (x)$ relations of trigonometry. How can I get the additon ...